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Math Help - Two questions: hypothesis test & how to win the car?

  1. #1
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    Red face Two questions: hypothesis test & how to win the car?

    Hi guys, I really need you help me with the following two problems:

    1. A drug trial gives the result that the drug works better than the placebo, with 95% confidence. What exactly does this statement mean? What further assumptions are needed to be able to deduce that the probability of the drug working is actually 95%?

    2. A company has a competition to win a car. Each contestant needs to pick a positive integer. If there's at least one uinique choice, the person who made the smallest unique choice wins the car. If there are no unique choices, the company keeps the car and there is no repeat of the competition. It turns out that there are only three contestants, and you are one of them. Everyone knows before picking their numbers that there are only three contestants. How should you make your choice?

    Any ideas are welcome! Many thanks.
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  2. #2
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    1) Generally, 95% confidence indicates that the mean effect of the drug is larger than the mean effect of the placebo 95% of the time over many repetitions.

    Assumptions are: what distribution is used to calculate confidence, the sample size, width of interval, is the control group independent of the drug group or is it a longitudinal study, is there any regression toward the mean, etc.

    2) This question could be difficult or easy. I would answer this question assuming that the other players pick a number 1, 2, or 3 at random.

    Assuming random picks, there are 27 combinations of (a, b, c) where a, b, c are the players and a, b, and c can take on 1, 2, or 3. If you list all combinations, you will win 8/27 times if you pick completely at random, but 4 of those wins will be if you pick the number 1. As, there are 9 combinations of picks when you choose 1, you win 4/9 of the time if you pick 1, 2/9 each player 2 and player 3 will win, and the company will win if you all pick 1.

    Thus, I would pick 1. Of course, all players, if intelligent would know that 1 is the best, so there could be some psychological game theory considerations, but I don't think that is what the question is asking.

    Hope this helps.

    [This question took 10 minutes to answer]
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  3. #3
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    Re: Two questions: hypothesis test & how to win the car?

    Can anyone shed some light on the below:

    1. Consider a set with N distinct members, and a function f defined on Q that takes the values 0, 1 such that (1/N)*Sum(over xEQ) of f(x) = p. For a subset S of Q of size n, define the sample proportion

    p := p(S) = (1/N)*Sum(over xES) of f(x)

    If each subset of size n is chosen with equal probability, calculate the expectation and
    standard deviation of the random variable p.

    2.
    (a) Let X-N(0, 1) be a normally distributed random variable with mean 0 and
    variance 1. Suppose that x E R, x > 0. Find upper and lower bounds for the conditional expectation

    E(X | X >x)

    (b) Now suppose that X has a power law distribution, P(X >x) = a *(x)^-b , for x>x0>0, and some a> 0, b> 1. Calculate the conditional expectation
    E(X|X>x), x >x0

    Many thanks in advance.
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